| bgcolor=#e7dcc3 colspan=2 | Parallelepiped | |
|---|---|---|
| align=center colspan=2 | ||
| Type | Prism Plesiohedron | |
| Faces | 6 parallelograms | |
| Edges | 12 | |
| Vertices | 8 | |
| Symmetry group | Ci, [2<sup>+</sup>,2<sup>+</sup>], (×), order 2 | |
| Properties | convex, zonohedron |
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square.
Three equivalent definitions of parallelepiped are
The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped.
"Parallelepiped" is now usually pronounced or ; traditionally it was [1] despite its etymology in Greek παραλληλεπίπεδον parallelepipedon, a body "having parallel planes".
Parallelepipeds are a subclass of the prismatoids.
Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four parallel edges; the edges within each set are of equal length.
Parallelepipeds result from linear transformations of a cube (for the non-degenerate cases: the bijective linear transformations).
Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry (see also triclinic). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not.
A space-filling tessellation is possible with congruent copies of any parallelepiped.
A parallelepiped is a prism with a parallelogram as base.Hence the volume
V
B
h
B=\left|a\right| ⋅ \left|b\right| ⋅ \sin\gamma=\left|a x b\right|
\gamma
a
b
h=\left|c\right| ⋅ \left|\cos\theta\right|
\theta
c
a=(a1,a2,a
| T, | |
| 3) |
~b=(b1,b2,b
| T, | |
| 3) |
~c=(c1,c2,c
| T, | |
| 3) |
Another way to prove is to use the scalar component in the direction of
a x b
c
An alternative representation of the volume uses geometric properties (angles and edge lengths) only:where
\alpha=\angle(b,c)
\beta=\angle(a,c)
\gamma=\angle(a,b)
a,b,c
The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see proof).
The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms:(For labeling: see previous section.)
| Form | Cube | Square cuboid | Trigonal trapezohedron | Rectangular cuboid | Right rhombic prism | Right parallelogrammic prism | Oblique rhombic prism | |
|---|---|---|---|---|---|---|---|---|
| Constraints | a=b=c \alpha=\beta=\gamma=90\circ | a=b \alpha=\beta=\gamma=90\circ | a=b=c \alpha=\beta=\gamma | \alpha=\beta=\gamma=90\circ | a=b \alpha=\beta=90\circ | \alpha=\beta=90\circ | a=b \alpha=\beta | |
| Symmetry | Oh order 48 | D4h order 16 | D3d order 12 | D2h order 8 | C2h order 4 | |||
| Image | ||||||||
| Faces | 6 squares | 2 squares, 4 rectangles | 6 rhombi | 6 rectangles | 4 rectangles, 2 rhombi | 4 rectangles, 2 parallelograms | 2 rhombi, 4 parallelograms | |
it has six rectangular faces (also called a rectangular parallelepiped, or sometimes simply a cuboid).
Note: the fully rhombic special case, with two rhombic faces and four congruent square faces
(a=b=c)
A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and space diagonals. In 2009, dozens of perfect parallelepipeds were shown to exist,[2] answering an open question of Richard Guy. One example has edges 271, 106, and 103, minor face diagonals 101, 266, and 255, major face diagonals 183, 312, and 323, and space diagonals 374, 300, 278, and 272.
Some perfect parallelepipeds having two rectangular faces are known. But it is not known whether there exist any with all faces rectangular; such a case would be called a perfect cuboid.
Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope. In modern literature, the term parallelepiped is often used in higher (or arbitrary finite) dimensions as well.[3]
Specifically in n-dimensional space it is called n-dimensional parallelotope, or simply -parallelotope (or -parallelepiped). Thus a parallelogram is a 2-parallelotope and a parallelepiped is a 3-parallelotope.
The diagonals of an n-parallelotope intersect at one point and are bisected by this point. Inversion in this point leaves the n-parallelotope unchanged. See also Fixed points of isometry groups in Euclidean space.
(v1,\ldots,vn)
The n-volume of an n-parallelotope embedded in
\Rm
m\geqn
If, this amounts to the absolute value of the determinant of matrix formed by the components of the vectors.
A formula to compute the volume of an -parallelotope in
\Rn
V0,V1,\ldots,Vn
[Vi 1]
Vi
Similarly, the volume of any n-simplex that shares n converging edges of a parallelotope has a volume equal to one 1/n! of the volume of that parallelotope.
The term parallelepiped stems from Ancient Greek (parallēlepípedon, "body with parallel plane surfaces"), from parallēl ("parallel") + epípedon ("plane surface"), from epí- ("on") + pedon ("ground"). Thus the faces of a parallelepiped are planar, with opposite faces being parallel.[4] [5]
In English, the term parallelipipedon is attested in a 1570 translation of Euclid's Elements by Henry Billingsley. The spelling parallelepipedum is used in the 1644 edition of Pierre Hérigone's Cursus mathematicus. In 1663, the present-day parallelepiped is attested in Walter Charleton's Chorea gigantum.
Charles Hutton's Dictionary (1795) shows parallelopiped and parallelopipedon, showing the influence of the combining form parallelo-, as if the second element were pipedon rather than epipedon. Noah Webster (1806) includes the spelling parallelopiped. The 1989 edition of the Oxford English Dictionary describes parallelopiped (and parallelipiped) explicitly as incorrect forms, but these are listed without comment in the 2004 edition, and only pronunciations with the emphasis on the fifth syllable pi (pronounced as //paɪ//) are given.